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G = C2×C40⋊C2order 160 = 25·5

Direct product of C2 and C40⋊C2

direct product, metabelian, supersoluble, monomial, 2-hyperelementary

Aliases: C2×C40⋊C2, C88D10, C4.6D20, C409C22, C101SD16, C20.29D4, C20.28C23, D20.6C22, C22.12D20, Dic103C22, (C2×C8)⋊5D5, (C2×C40)⋊7C2, C51(C2×SD16), C10.9(C2×D4), (C2×D20).5C2, C2.11(C2×D20), (C2×C10).16D4, (C2×C4).79D10, (C2×Dic10)⋊5C2, C4.26(C22×D5), (C2×C20).88C22, SmallGroup(160,123)

Series: Derived Chief Lower central Upper central

C1C20 — C2×C40⋊C2
C1C5C10C20D20C2×D20 — C2×C40⋊C2
C5C10C20 — C2×C40⋊C2
C1C22C2×C4C2×C8

Generators and relations for C2×C40⋊C2
 G = < a,b,c | a2=b40=c2=1, ab=ba, ac=ca, cbc=b19 >

Subgroups: 280 in 68 conjugacy classes, 33 normal (17 characteristic)
C1, C2, C2 [×2], C2 [×2], C4 [×2], C4 [×2], C22, C22 [×4], C5, C8 [×2], C2×C4, C2×C4, D4 [×3], Q8 [×3], C23, D5 [×2], C10, C10 [×2], C2×C8, SD16 [×4], C2×D4, C2×Q8, Dic5 [×2], C20 [×2], D10 [×4], C2×C10, C2×SD16, C40 [×2], Dic10 [×2], Dic10, D20 [×2], D20, C2×Dic5, C2×C20, C22×D5, C40⋊C2 [×4], C2×C40, C2×Dic10, C2×D20, C2×C40⋊C2
Quotients: C1, C2 [×7], C22 [×7], D4 [×2], C23, D5, SD16 [×2], C2×D4, D10 [×3], C2×SD16, D20 [×2], C22×D5, C40⋊C2 [×2], C2×D20, C2×C40⋊C2

Smallest permutation representation of C2×C40⋊C2
On 80 points
Generators in S80
(1 77)(2 78)(3 79)(4 80)(5 41)(6 42)(7 43)(8 44)(9 45)(10 46)(11 47)(12 48)(13 49)(14 50)(15 51)(16 52)(17 53)(18 54)(19 55)(20 56)(21 57)(22 58)(23 59)(24 60)(25 61)(26 62)(27 63)(28 64)(29 65)(30 66)(31 67)(32 68)(33 69)(34 70)(35 71)(36 72)(37 73)(38 74)(39 75)(40 76)
(1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40)(41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80)
(1 77)(2 56)(3 75)(4 54)(5 73)(6 52)(7 71)(8 50)(9 69)(10 48)(11 67)(12 46)(13 65)(14 44)(15 63)(16 42)(17 61)(18 80)(19 59)(20 78)(21 57)(22 76)(23 55)(24 74)(25 53)(26 72)(27 51)(28 70)(29 49)(30 68)(31 47)(32 66)(33 45)(34 64)(35 43)(36 62)(37 41)(38 60)(39 79)(40 58)

G:=sub<Sym(80)| (1,77)(2,78)(3,79)(4,80)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,77)(2,56)(3,75)(4,54)(5,73)(6,52)(7,71)(8,50)(9,69)(10,48)(11,67)(12,46)(13,65)(14,44)(15,63)(16,42)(17,61)(18,80)(19,59)(20,78)(21,57)(22,76)(23,55)(24,74)(25,53)(26,72)(27,51)(28,70)(29,49)(30,68)(31,47)(32,66)(33,45)(34,64)(35,43)(36,62)(37,41)(38,60)(39,79)(40,58)>;

G:=Group( (1,77)(2,78)(3,79)(4,80)(5,41)(6,42)(7,43)(8,44)(9,45)(10,46)(11,47)(12,48)(13,49)(14,50)(15,51)(16,52)(17,53)(18,54)(19,55)(20,56)(21,57)(22,58)(23,59)(24,60)(25,61)(26,62)(27,63)(28,64)(29,65)(30,66)(31,67)(32,68)(33,69)(34,70)(35,71)(36,72)(37,73)(38,74)(39,75)(40,76), (1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40)(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80), (1,77)(2,56)(3,75)(4,54)(5,73)(6,52)(7,71)(8,50)(9,69)(10,48)(11,67)(12,46)(13,65)(14,44)(15,63)(16,42)(17,61)(18,80)(19,59)(20,78)(21,57)(22,76)(23,55)(24,74)(25,53)(26,72)(27,51)(28,70)(29,49)(30,68)(31,47)(32,66)(33,45)(34,64)(35,43)(36,62)(37,41)(38,60)(39,79)(40,58) );

G=PermutationGroup([(1,77),(2,78),(3,79),(4,80),(5,41),(6,42),(7,43),(8,44),(9,45),(10,46),(11,47),(12,48),(13,49),(14,50),(15,51),(16,52),(17,53),(18,54),(19,55),(20,56),(21,57),(22,58),(23,59),(24,60),(25,61),(26,62),(27,63),(28,64),(29,65),(30,66),(31,67),(32,68),(33,69),(34,70),(35,71),(36,72),(37,73),(38,74),(39,75),(40,76)], [(1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,36,37,38,39,40),(41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80)], [(1,77),(2,56),(3,75),(4,54),(5,73),(6,52),(7,71),(8,50),(9,69),(10,48),(11,67),(12,46),(13,65),(14,44),(15,63),(16,42),(17,61),(18,80),(19,59),(20,78),(21,57),(22,76),(23,55),(24,74),(25,53),(26,72),(27,51),(28,70),(29,49),(30,68),(31,47),(32,66),(33,45),(34,64),(35,43),(36,62),(37,41),(38,60),(39,79),(40,58)])

C2×C40⋊C2 is a maximal subgroup of
C85D20  C8.8D20  C42.16D10  C8⋊D20  C8.D20  D20.31D4  D20.32D4  D2014D4  Dic1014D4  Dic102D4  D20.8D4  D43D20  D20.D4  Dic10.11D4  Q82D20  Q8.D20  Dic5⋊SD16  C20⋊SD16  D20.19D4  C42.36D10  Dic108D4  Dic58SD16  C88D20  C83D20  C4021(C2×C4)  C8.24D20  C4030D4  C402D4  D4.3D20  C4011D4  C40.43D4  C4015D4  C40.37D4  D4.11D20  C2×D5×SD16  D811D10
C2×C40⋊C2 is a maximal quotient of
C409Q8  C20.14Q16  C85D20  C4.5D40  C23.34D20  D20.31D4  C23.38D20  Dic1014D4  C20⋊SD16  D203Q8  Dic108D4  Dic104Q8  C4030D4

46 conjugacy classes

class 1 2A2B2C2D2E4A4B4C4D5A5B8A8B8C8D10A···10F20A···20H40A···40P
order122222444455888810···1020···2040···40
size111120202220202222222···22···22···2

46 irreducible representations

dim11111222222222
type++++++++++++
imageC1C2C2C2C2D4D4D5SD16D10D10D20D20C40⋊C2
kernelC2×C40⋊C2C40⋊C2C2×C40C2×Dic10C2×D20C20C2×C10C2×C8C10C8C2×C4C4C22C2
# reps141111124424416

Matrix representation of C2×C40⋊C2 in GL3(𝔽41) generated by

4000
010
001
,
100
02728
01318
,
100
0134
0040
G:=sub<GL(3,GF(41))| [40,0,0,0,1,0,0,0,1],[1,0,0,0,27,13,0,28,18],[1,0,0,0,1,0,0,34,40] >;

C2×C40⋊C2 in GAP, Magma, Sage, TeX

C_2\times C_{40}\rtimes C_2
% in TeX

G:=Group("C2xC40:C2");
// GroupNames label

G:=SmallGroup(160,123);
// by ID

G=gap.SmallGroup(160,123);
# by ID

G:=PCGroup([6,-2,-2,-2,-2,-2,-5,218,50,579,69,4613]);
// Polycyclic

G:=Group<a,b,c|a^2=b^40=c^2=1,a*b=b*a,a*c=c*a,c*b*c=b^19>;
// generators/relations

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